Does $\int _{\mathbb{R}}f_{j} dm \rightarrow 0$ imply $f_{j} \rightarrow 0$?

Question

Let $f_{j}$ be a sequence of non-negative Lebesgue measured functions $f_{j}: \mathbb{R} \rightarrow \mathbb{R}$ s.t.

$\int _{\mathbb{R}}f_{j} dm \rightarrow 0$.

Does now hold $f_{j} \rightarrow 0$ for almost everywhere $x \in \mathbb{R}$?

Answer 1

Consider the sequence \begin{align} f_{1,2}(x)&=\chi_{[0,1/2]}(x) \\ f_{2,2}(x)&=\chi_{[1/2,1]}(x) \\ f_{1,3}(x)&=\chi_{[0,1/3]}(x) \\ f_{2,3}(x)&=\chi_{[1/3,2/3]}(x) \\ f_{3,3}(x)&=\chi_{[2/3,1]}(x)\\ \vdots \end{align} The sequence of integrals goes to $0$ but the sequence does not converge to $0$ pointwise anywhere in $[0,1]$.